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Properties

Label 213.5.d.a
Level $213$
Weight $5$
Character orbit 213.d
Analytic conductor $22.018$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,5,Mod(70,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.70");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 213.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.0178021369\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} + 372 q^{4} - 552 q^{8} + 1296 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} + 372 q^{4} - 552 q^{8} + 1296 q^{9} + 244 q^{10} + 360 q^{12} - 252 q^{15} + 4140 q^{16} - 324 q^{18} - 88 q^{19} - 444 q^{20} + 7116 q^{25} + 1548 q^{29} + 2664 q^{30} - 5064 q^{32} + 10044 q^{36} - 1324 q^{37} + 17568 q^{38} - 1284 q^{40} - 10036 q^{43} + 9936 q^{48} - 18256 q^{49} - 20496 q^{50} - 2880 q^{57} + 3080 q^{58} - 1044 q^{60} + 6840 q^{64} - 19176 q^{71} - 14904 q^{72} + 5704 q^{73} + 15924 q^{74} - 8712 q^{75} + 2924 q^{76} - 12036 q^{77} + 236 q^{79} - 5460 q^{80} + 34992 q^{81} - 18252 q^{83} - 4788 q^{86} + 3744 q^{87} - 37620 q^{89} + 6588 q^{90} - 35148 q^{91} - 60360 q^{95} + 9900 q^{96} + 55188 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −7.65394 5.19615 42.5828 19.9591 −39.7710 74.6745i −203.463 27.0000 −152.766
70.2 −7.65394 5.19615 42.5828 19.9591 −39.7710 − 74.6745i −203.463 27.0000 −152.766
70.3 −7.15057 −5.19615 35.1306 33.5628 37.1555 53.5034i −136.795 27.0000 −239.993
70.4 −7.15057 −5.19615 35.1306 33.5628 37.1555 − 53.5034i −136.795 27.0000 −239.993
70.5 −7.11941 −5.19615 34.6860 −20.3958 36.9935 4.66897i −133.033 27.0000 145.206
70.6 −7.11941 −5.19615 34.6860 −20.3958 36.9935 − 4.66897i −133.033 27.0000 145.206
70.7 −6.94977 5.19615 32.2993 −48.1070 −36.1121 − 91.1294i −113.277 27.0000 334.333
70.8 −6.94977 5.19615 32.2993 −48.1070 −36.1121 91.1294i −113.277 27.0000 334.333
70.9 −6.04986 5.19615 20.6009 4.55202 −31.4360 − 5.58839i −27.8346 27.0000 −27.5391
70.10 −6.04986 5.19615 20.6009 4.55202 −31.4360 5.58839i −27.8346 27.0000 −27.5391
70.11 −4.97027 −5.19615 8.70357 −27.3183 25.8263 64.9583i 36.2652 27.0000 135.779
70.12 −4.97027 −5.19615 8.70357 −27.3183 25.8263 − 64.9583i 36.2652 27.0000 135.779
70.13 −4.14118 −5.19615 1.14938 25.3412 21.5182 3.51374i 61.4991 27.0000 −104.942
70.14 −4.14118 −5.19615 1.14938 25.3412 21.5182 − 3.51374i 61.4991 27.0000 −104.942
70.15 −3.44841 5.19615 −4.10844 33.7793 −17.9185 68.9607i 69.3422 27.0000 −116.485
70.16 −3.44841 5.19615 −4.10844 33.7793 −17.9185 − 68.9607i 69.3422 27.0000 −116.485
70.17 −3.32359 5.19615 −4.95372 −22.7817 −17.2699 15.2343i 69.6417 27.0000 75.7170
70.18 −3.32359 5.19615 −4.95372 −22.7817 −17.2699 − 15.2343i 69.6417 27.0000 75.7170
70.19 −1.61844 −5.19615 −13.3807 −28.5122 8.40965 − 57.5234i 47.5508 27.0000 46.1452
70.20 −1.61844 −5.19615 −13.3807 −28.5122 8.40965 57.5234i 47.5508 27.0000 46.1452
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.5.d.a ✓ 48
71.b odd 2 1 inner 213.5.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.5.d.a ✓ 48 1.a even 1 1 trivial
213.5.d.a ✓ 48 71.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(213, [\chi])\).